Communications in Mathematical Physics

, Volume 78, Issue 4, pp 455–478 | Cite as

Symmetry and bifurcations of momentum mappings

  • Judith M. Arms
  • Jerrold E. Marsden
  • Vincent Moncrief


The zero set of a momentum mapping is shown to have a singularity at each point with symmetry. The zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function. The proof uses the Kuranishi theory of deformations. Among the applications, it is shown that the set of all solutions of the Yang-Mills equations on a Lorentz manifold has a singularity at any solution with symmetry, in the sense of a pure gauge symmetry. Similarly, the set of solutions of Einstein's equations has a singularity at any solution that has spacelike Killing fields, provided the spacetime has a compact Cauchy surface.

Glossary of Symbols


symplectic manifold

\(T_{x_0 } P\)

tangent space toP atx0εP


Lie group, Lie algebra


action ofG onP


infinitesimal generator of the action onP corresponding toξεg


momentum mapping

\(dJ\left( {x_0 } \right):T_{x_0 } P \to \mathfrak{g}^ * \)

differential ofJ atx0


complex structure onP

\(S_{x_0 } \)

slice for theG-action atx0

\(I_{x_0 } \)

isotropy group ofx0; {g εG|gx0=x0}

\(\mathcal{L}_{x_0 } ,\mathfrak{s}_{x_0 } \)

identity component of\(I_{x_0 } \), its Lie algebra [Eq. (10)]


(weak) metric onP


pairing between g* and g

\((,)_{x_0 } \)

inner product of g* depending onx0εP

\(dJ\left( {x_0 } \right)^ * :\mathfrak{g} \to T_{x_0 } P\)

adjoint ofdJ(x0) relative to 〈,〉 and «,»

\(dJ\left( {x_0 } \right)^\dag :\mathfrak{g}^ * \to T_{x_0 } P\)

adjoint ofdJ(x0) relative to (,) and «,»

\(T_{x_0 } P = Range\left[ {\mathbb{J} \circ dJ\left( {x_0 } \right)^ * } \right] \oplus Range\left[ {dJ\left( {x_0 } \right)^ * } \right] \oplus \left[ {\ker \left( {dJ\left( {x_0 } \right) \circ \mathbb{J}} \right) \cap \ker dJ\left( {x_0 } \right)} \right]\)

Moncrief's decomposition


orthogonal projection

\(\mathcal{C} = J^{ - 1} \left( 0 \right)\)

zero set ofJ (or constraint set)

\(\mathcal{C}_\mathbb{P} = \left( {\mathbb{P}J} \right)^{ - 1} \left( 0 \right)\)

zero set of ℙ∘J

\(N_{x_0 } \)

x's with the same orbit type asx0

\(\mathcal{N}_{x_0 } \)

\(N_{x_0 } \cap S_{x_0 } \) (Lemma 1)

\(\mathfrak{g}_{x_0 }^ * \)

elements in g* with same symmetry asx0 (Lemma 5)


\(f = \left( {Id - \mathbb{P}} \right) \circ J:\mathcal{C}_\mathbb{P} \cap S_{x_0 } \to \ker dJ\left( {x_0 } \right)^\dag \) (Lemma 8)


“elliptic” operator associated withJ


Greens' function for Δ


Kuranishi map (Lemma 9)

\(C_{x_0 } \)

homogeneous cone associated withd2J(x0) (Theorem 3)

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orthogonal projection onto kerdJ(x0) (Theorem 3)

\(\mathfrak{h}\), ℋ

a Lie subalgebra of\(\mathfrak{s}_{x_0 } \), its Lie group

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points with symmetry (at least) ℋ (Theorem 4)

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Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • Judith M. Arms
    • 1
  • Jerrold E. Marsden
    • 2
  • Vincent Moncrief
    • 3
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  3. 3.Department of PhysicsYale UniversityNew HavenUSA

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